# Simple Geometry: find point from various lines

In the following diagram, everything occurs in a plane. AB and BC are line segments. DE is a line parallel to AB, and FG is a line parallel to BC. The distance between AB & DE is equal to the distance between BC & FG. Given the points A, B, and C, and the distance between AB & DE, what is the most efficient way to find the location of point I, the intersection of DE and FG? My initial approach was to calculate two points on the line DE, two points on the line FG, and find the intersection point of the two lines through those points. However, I was thinking, would the line HJ which passes though both B and I always cut the angle ABC exactly in half? If so, then perhaps a faster method of finding I is to calculate the angle ABI, the distance BI, and find I relative to B.

My geometry is a little rusty: although it seems to me like angle ABI should always be one half of angle ABC, I don't want to assume that without a little more rigor.

My question is: Does angle ABI always equal half of angle ABC, and/or, do you see a more efficient way of determining the location of point I given A, B, C, and the distance between AB & DE? Therefore, $\angle BIH = \angle BIK$ [corresponding measurements]
Thus, $\angle ABJ = \angle BIH = \angle BIK = \angle CBJ$ [corresponding angles of parallel lines]